A simple method for estimatingMSYfrom catch and resilience

Running title: Catch-MSY

Steven Martell1 and Rainer Froese2*

1University of British Columbia, 2202 Main Mall, Vancouver, BC, V6T 1Z4

2*Helmholtz Centre for Ocean Research, GEOMAR, Düsternbroker Weg 20, 24105, Kiel, Germany, Tel. +49 431 600 4579, Fax +49 431 600 1699, email , corresponding author

Abstract

The Law of the Sea requires that fish stocks are maintained at levels that can produce the maximum sustainable yield (MSY). However, for most fish stocks no estimates of MSY are currently available. Here we present a new method for estimating MSY from catch data, resilience of the respective species, and simple assumptions about relative stock sizes at the first and final year of the catch data time series. We compare our results with 146 MSY estimates derived from full stock assessments and find excellent agreement. We present principles for fisheries management of data-poor stocks, based only on information about catches and MSY.

Keywords

data-poor stocks, maximum sustainable yield, intrinsic rate of population increase, carrying capacity, harvest control rules

Contents

Introduction

The need for simple methods

Outline of the Catch-MSY method

Material and Methods

Model and assumptions

Data sources

Results and Discussion

Applying the Catch-MSY method to Greenland halibut

Applying the Catch-MSY method to other stocks with available MSY estimates

How good are the estimates of r and k?

Applying the Catch-MSY method to Strait of Georgia lingcod

Caveats of the Catch-MSY approach

Other methods for estimatingMSY from catch data

Principles of catch & MSY management

Summary

Acknowledgements

References

Appendix

Introduction

The need for simple methods

In the Law of the Sea of 1982 (UNCLOS 1982), which entered into force in 1994, the nations of the World have agreed to maintain exploited populations of marine organisms at levels that can produce the maximum sustainable yieldMSY. Respectivemanagement systems have been introduced by Australia (DAFF 2007), New Zealand (MFNZ 2008) and the USA (MSA 2006) and Europe plans an implementation by 2013 (EC 2011). However, for the vast majority of exploited populations or stocks,no estimates of MSY are available. Thus, there is a need for simple methods that allow inclusion of such stocks in MSY management schemes.

Outline of the Catch-MSY method

The simplest model-based methods for estimating MSY are production models such as the Schaefer model (1954). At a minimum these models require time series data of abundance and removals to estimate two model parameters: the carrying capacity kand the maximum rate of population increase rfor a given stock in a given ecosystem. While estimates of removals (defined here as catch plus dead discards) are available for most stocks, abundance estimates are difficult and costly to obtain and are mostly missing. However, given only a time series of removals, a surprisingly narrow range of r-kcombinations is able to maintain the population such that it neither collapses nor exceeds the assumed carrying capacity. This set of viabler-kcombinations can be used to approximateMSY. Here we present a simple method that uses catch data plus readily available additional information to approximate MSY with error margins. We demonstrate the application for two stocks, Greenland halibut (Reinhardtius hippoglossoides, Pleuronectidae) andStrait of Georgia lingcod (Ophiodon elongatus, Hexagrammidae).We apply the method to 48 stocks of the Northeast Atlantic for which independent MSY estimates were available from a previous study and compare the results. We also apply the method to 98 global stocks with MSY estimates. Finally, afterpointing out some caveats, we propose simple management rules based on catch and MSY.

Material and Methods

Model and assumptions

The Catch-MSY methodas proposed here was inspired by the stock reduction analysis of Kimura and Tagart (1982) and Kimura et al. (1984). As input data it requires a time series of removals, prior ranges of r and k, and possible ranges of relative stock sizes in the first and final years of the time series.It then uses the Schaefer production model to calculate annual biomasses for a given set of r and k parameters.Since no prior distributions of r and k are available for most fish stocks, we randomly draw r-k pairs from a uniform prior distribution and then use a Bernoulli distribution as the likelihood function for accepting each r-k pair that has never collapsed the stock or exceeded carrying capacity, and that results in a final relative biomass estimate that falls within the assumed range of depletion.Additional process errors can also be added to the model if desired. Absent process errors, as in our examples, is equivalent to assuming an observation error only model that is deterministic. A detailed description of the parameters and equations is given in the Appendix.The R-code for batch processing of the 146 stocks and the catch data are available from with file names of CatchMSY_2.r, RAM_MSY.csv, andICESct2.csv concatenated to the URL, respectively.

Data sources

We used assessment data for 48 stocks of 19 species of the Northeast Atlantic, as available in the ICES Stock Summary Database downloaded from in September 2011. We extracted estimates of F0.1 from ICES advice documents for 2011, as available from We also used the estimates of Fmsy, MSY, and carrying capacity k for these stocks from Froese and Proelss (2010). For each species we got a resilience classification from FishBase (Froese and Pauly 2011).These stocks spanned a wide range of sizes and exploitation rates, ranging in spawning stock biomass from 1,000 tonnes to 12 million tonnes, with exploitation rates F/Fmsy of 0.5 to 5.8. The advantage of this data set were the application of the same standard methods across all stocks, and the provision ofMSY with 95% confidence limits by Froese and Proelss (2010). The disadvantage of this data set was that, with one exception, it only contained species with medium resilience.We therefore also used working group assessments of MSY for 98 stocks from the RAM legacy database (Ricard et al. 2011). For the batch analysis of these stocks we derived default ranges of relative biomass in the first and final year of the time series, based on respective catches relative to the maximum catch (Froese et al. 2012), see Table 1.

Random samples of the carrying capacity parameter (k)were drawn from a uniform distribution where the lower and upper limits were given by the maximum catch in the time series and 100 times maximum catch, respectively. Note that such upper bound for k means that catches never exceeded 1% of the carrying capacity. If this were indeed the case, catches would contain very little information about the productivity of the stock and the Catch-MSY method should not be applied. Given their nearly unexploited status, such stocks are not in immediate need of management.

We used resilience estimates from FishBase, which are based on Musick (1999) as modified by Froese et al. (2000), to assign default values to the allowed range for the random samples of themaximum intrinsic rate of population increase r(Table 2). Note that we do not propose application of the Catch-MSY method with the default values in Tables 1 and 2 for serious stock assessment. Rather, we would expect that the best available knowledge about the respective stocks is used.

As most probable values from the resulting density distributions we used the geometric means of r, k, and MSY, where MSY wascalculated from ther-kpairs (see Appendix).We chose geometric mean instead of mean, median or mode because it was the only estimate where the central MSYvalue derived after calculation of MSY for each r-k pair was about the same as the one derived by using the respective central values of r and k. For example, for Western Baltic cod, median MSY calculated from r-k pairs was 38,335 tonnes, whereas MSY calculated from median r and k was 38,997 tonnes, a difference of 662 tonnes. For the geometric mean the respective values were 38,975 and 38,906, a difference of only 69 tonnes. Thus, the geometric mean seemed to better capture the distributions of r, k, and MSY.

As measure of uncertainty we used two times the standard deviation of the logarithmic mean. This implies that, with a roughly log-normal distribution, about 95% of the MSY estimates would fall within this range.

Results and Discussion

Applying the Catch-MSY method to Greenland halibut

Figure 1 shows the graphical output of the Catch-MSY method as applied to the Greenland halibut, a species with low resilience (Froese and Pauly 2011, see Table 1). Panel A shows the time series of catches with overlaid estimate of MSY= 24,900 tonnesand the limits (19,800 – 31,400) that contain about 95% of the MSYestimates derived from the r-k pairs. This is not significantly different from an independent estimate for this stock of MSY = 31,023 tonnes with 95% confidence limits of 19,171 – 53,950 tonnes (Froese and Proelss 2010). Panel B spans the prior uniform distribution of r= 0.05 - 0.5 and k= 89,484 – 8,948,400 tonnes.The r-k combinations (1st iterations) that are compatible with the time series of catches occupy only a small corner of that space, showing the typical decline of viable r-kpairs with increasing r. Panel C is a magnification of the r-k pairs (after 2nd iterations with new upper limit for k) in log space, with overlaid lines indicating the r-k combinations that would result in geometric mean MSY +/- 2 standard deviations. Panels D to F show the posterior densities of r, k and MSY, respectively.

Applying the Catch-MSY method toother stocks with available MSY estimates

The key question obviously is how well the MSY estimates derived with the Catch-MSY method compare with a wide range ofMSYestimates from full stock assessments. For this comparison we used 48 stocks from the Northeast Atlanticand 98 stocks from all over the world, analyzed with the default assumptions as described above. These default settings found r-kpairs for all Northeast Atlantic stocks and for mostof the global stocks. Figure 2 shows a comparison of the respective MSY estimates for the 48 Northeast Atlantic stocks. Alog-log linear regression accounted for 98.6% of the variability of Catch-MSY estimates relative to full assessment estimates of MSY, with an intercept not significantly different from the origin (n = 48, log intercept = -0.05, 95% CL = -0.118 – 0.018) and a slope not significantly different from 1 (slope = 1.003, CL = 0.967 – 1.039, r2 = 0.986). The 95% confidence limits of MSY provided by Froese & Proelss (2010) overlapped in 42 of the 48 stocks with the double standard deviation used as error margin by the Catch-MSY method, suggesting that these MSY estimates were not significantly different.

For the global stocks, the default settings did not result in suitable r-k combinations for about 10 of the 98 stocks, mostly because these stocks had intermediate resilience (between Very low and Low or between Low and Medium, see Table 2), or because they were very lightly exploited, with maximum catches of 2 - 30% of the MSY estimate of the respective working groups, see outliers in Figure 3. As pointed out above, in very lightly fished stocks the time series of catches does not contain sufficient information about productivity and the Catch-MSY method should not be applied. But overall, most of the Catch-MSY estimates for the global stocks fell within a range of 0.5 to 1.5 of the independent estimate, see respective lines in Figure 3. Thus, the Catch-MSY methodappears well suited to provide preliminary approximations of MSY in cases where abundance data are lacking.

How good are the estimates of r and k?

A stockthat was able to produce the cumulative historical catchmust have had a certain productivity,for which the maximum sustainable yield is an appropriate measure. MSY is a function of r and k, whichin the Schaefer model have a log-linear negative correlation with a slope of -1 (Figure 1C). In other words, the observed catches may have been produced by a small population with high r or a large population with small r, or a very lightly exploited population with any combination of r and k.In the last case, catch data are insufficient to estimate population properties, error margins for r, k and MSY will be very wide, and MSY will be underestimated (outliers in Figure 3). High values ofrcause strong fluctuations in stock size, with the associated risks of overshooting carrying capacityor going extinct (May 1974, 1976). Also, the larger k is relative to catches, the wider is the range of r values that allow the population to sustain those catches. These two effects may explain the typical triangular shape of viabler-kpairsinlogr-k space(Figure 1 C), with few pairs at high and most pairs at lowerrvalues.This triangle only expands by its short side, when the range for prior r is reduced and for priork increased.

The triangular shape of viable r-k spacedoes not affect much the estimation of a representative value forMSY, as the line ofr-k pairs giving the same MSY is anchored near the center of the triangle(Fig. 1C). However, the estimates of most probable central values for r and kare strongly dependent on the lower limit chosen for r and the upper limit chosen for k. While the lower limit for r is assumed to represent the best available prior knowledge, the upper limit for k was chosen arbitrarily as 100-fold the maximum catch in the time series. We used the following method for finding an upper limit of kbetter correspondingwith the lower limit of r, based on the knowledge gained in a first analysis of the data:We selected as upper limit of kthe smallest viablekvalue at thelower limit of r. This provided a clear new upper cut-off for k determined by the prior lower limit of r (Fig. 1C).

In Figures 4-6 we compare the r and k estimates of the Catch-MSY method with related fisheries reference points. Figure 4 shows a plot of k over unexploited total biomass from Froese and Proelss(2010). The points scatter around the 1:1 line, but with an upward bias of about 10%, i.e., the Catch-MSY method overestimated carrying capacity and related biomass reference points by about that amount. Similarly, in Figure 5, most estimates of r fall below the 1:2 line of the relationship between r and the fishing mortality Fmsy that would result in the biomass that can produce maximum sustainable yield (data from Froese and Proelss 2010). A better match is obtained in Figure 6, where r is plotted over the conservative fishing mortality F0.1 derived by ICES working groups from yield per recruit analysis (Cadima 2003). The rectangular distribution of the data points in Figures 5 and 6 stems from the fact that most species were of medium resilience and thus had the same default lower prior limit of r = 0.2.

In summary, while MSY estimates of the Catch-MSY methodare fairly robust with regard to initial assumptions and in very good agreement with estimates derived with more demanding methods, the r and k estimates strongly depend on the lower prior limit for rwhich thus must be carefully set. From a management point of view, the bias of r and k is precautionary, because it suggests higher thresholds for biomass and lower thresholds for fishing mortality.

Applying the Catch-MSY methodto the ‘data-poor’Strait of Georgia lingcod

To demonstrate this simple method for estimating MSY from catch time series data, we used the historical landings from the Strait of Georgia lingcod fishery, for which no abundance data were available (Fig.7) (King 2001). Landings of lingcod from this region date back to 1889. The commercial fishery largely consisted of a handline fishery; however, lingcod were also taken in trawl fisheries, starting in the 1940s. The stock was considered depleted and the commercial fishery was closed in 1990. The remaining recreational fishery was closed in 2002 (Logan et al. 2005).

The lingcod fishery began around 1860 (King 2001) and thus we assumed that the biomass at the start of the time series in 1889 was already below carrying capacity. Since early catches were modest, we assumed a start biomass of 0.8k. For the prior densities ofr we took the uniform distribution for species with very low resiliencewith r = 0.015 - 0.1 (Table 1). Given the known depleted state of the stock, we assumed that relative biomass in 2002 was between 1% and 25% of the carrying capacity. Inserting the estimated MSY in Figure 7 facilitated interpretation: Landings reached the lower range of MSY estimates in the 1900s, exceededMSY in the 1910s and the upper range of MSY estimates between 1920 and 1960, and declined thereafter.Despite the closure of the commercial fishery in 1990, the stock showed no consistent signs of recovery in 2000 (Logan et al. 2005).A full assessment is still not available for this stock (DFO 2012), but Figure 7suggests that the massive and prolonged overshooting of MSY was the cause of depletion in 1990.

Caveats of the Catch-MSY approach

A key assumption in the Catch-MSY approach as laid out here is the ability to define a reasonable prior range for the parameters of the Schaefer model. In our case studies wehave arbitrarily chosen 100 times the maximum catch as the upper bound for k. In a developing fishery, or a fishery that has a continuous increase in catch, it will be more difficult to define the upper bound of k because the maximum potential has yet to be realized.Another key assumption is the stated range of depletion for which to accept or reject sets of r-k pairs of parameters. The lower depletion limit defines the lower boundary of the resulting MSY distribution and the upper depletion limit and the range of values for k determine the upper bound of MSY. To be clear, these depletion levels are assumptions about the current state of the stock. Finally, the Catch-MSY approach also assumes a stationary production function, or in this case no change in the parameters of the Schaefer model over time.